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Physical Review

Light Quanta and Wave Mechanics

J. C. Slater

Published May 1, 1928• 9 citations
Publisher

Abstract

Light quanta are treated by wave mechanics by analogy with electrons. It is shown that their wave equation is the ordinary optical wave equation. Heisenberg's principle of indeterminateness becomes a description of diffraction. In problems in which localization of quanta is found experimentally, wave packets are to be set up; these are applied to the experiments of Bothe and Geiger, and of Compton. The paths of quanta coincide with the rays of geometrical optics, with a deviation of the error in geometrical optics; by the principle of indeterminateness, more accurate laws for the paths are neither necessary nor possible.

DOI
10.1103/physrev.31.895

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Light quanta are treated by wave mechanics by analogy with electrons. It is shown that their wave equation is the ordinary optical wave equation. Heisenberg's principle of indeterminateness becomes a description of diffraction. In problems in which localization of quanta is found experimentally, wave packets are to be set up; these are applied to the experiments of Bothe and Geiger, and of Compton. The paths of quanta coincide with the rays of geometrical optics, with a deviation of the error in geometrical optics; by the principle of indeterminateness, more accurate laws for the paths are neither necessary nor possible.
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The five laws of relativistic quantum mechanics, according to Feynman's path integral formulation, are concisely stated and applied to experiments. Reflection diffraction grating experiments for both photons and electrons are analysed, in particular the Davisson-Germer experiment in which the wave-like property of electrons was first established. It is shown how classical, purely spatial, effective wave theories for both photons and electrons are predicted by the path integral formulation of quantum mechanics. The standard Copenhagen interpretation of wave mechanics is critically discussed in the light of the described experimental applications of the path integral formulation.
Let us return from optics to mechanics and explore the analogy to its fullest extent. In optics the old system of mechanics corresponds to intellectually operating with isolated mutually independent light rays. The new undulatory mechanics corresponds to the wave theory of light . Erwin Schrödinger, Nobel lecture , 1933 In this chapter we shall see that electromagnetic waves and electrons feature a number of common properties under conditions of spatial confinement. Simple and familiar problems from introductory quantum mechanics and textbook wave optics are recalled in this chapter to emphasize the basic features of waves in spatially inhomogeneous media. Herewith we make a first step towards understanding the properties of electrons and electromagnetic waves in nanostructures and notice that these properties in many instances are counterparts. Different formulas and statements of this chapter can be found in handbooks on quantum mechanics and wave optics. A few textbooks on quantum mechanics do consider analogies of propagation and reflection phenomena in wave optics with those in wave mechanics. Isomorphism of the Schrödinger and Helmholtz equations In Chapter 2 we discussed that an electron in quantum mechanics is described by the wave function, the square of its absolute value giving the probability of finding an electron at a specific point in space. This function satisfies the Schrödinger equation (2.56) which is the second-order differential equation with respect to space and the first-order differential equation with respect to time.
The problem of emission by an atom is treated according to Schroedinger’s rules. The translational motion of the atom as a whole is taken into account. The charge-current density vector is shown to be propagating with the velocity of light in the direction of the emitted quantum. Only unidirectional quanta can be emitted according to the theory. The reason is that the charge-current wave radiates infinitely more intensely if it travels with light velocity than otherwise.It is shown that Gordon’s treatment of the Compton effect involves an analogous phenomenon. It is suggested that Schroedinger’s equations designed for the conservation of energy and momentum work out better when the translational motion of the atom as a whole is considered.
It is shown that the relation between ray optics and wave optics is analogous to the relation between the classical mechanics of a point particle and quantum mechanics. The time coordinate of quantum mechanics has to be replaced with a length coordinate and the light wavelength λ takes the place of Planck’s constant h. A well-known result of quantum mechanics is that it coincides with classical mechanics in the limit h → 0. Correspondingly, wave and ray optics must coincide in the limit λ → 0.